Compact Non-Lie Groups

Abstract

The principal unsolved problem in this area is the famous Hilbert-Smith Conjecture: If a compact group G acts freely** on a manifold, then G is a Lie group. To prove this conjecture, it would suffice, [4, 8 or 1], to show that no p-adic group can act freely on a manifold. Thus, in the past, researchers have looked for whatever surprising consequences they could find, from the assumption that a p-adic group A p acts freely on an n-manifold X.